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One usually thinks of a radial density profile as having a monotonicallychanging logarithmic slope, such as in NFW or Einasto profiles. However, in twodifferent classes of commonly used systems, this is often not the case. Theseclasses exhibit non-monotonic changes in their density profile slopes which wecall oscillations for short. We analyze these two unrelated classes separately.Class 1 consists of systems that have density oscillations and that are definedthrough their distribution function $f(E)$, or differential energy distribution$N(E)$, such as isothermal spheres, King profiles, or DARKexp, a theoreticallyderived model for relaxed collisionless systems. Systems defined through $f(E)$or $N(E)$ generally have density slope oscillations. Class 1 systemoscillations can be found at small, intermediate, or large radii but we focuson a limited set of Class 1 systems that have oscillations in the centralregions, usually at $\log(r/r_{-2})\lesssim -2$, where $r_{-2}$ is the largestradius where $d\log(\rho)/d\log(r)=-2$. We show that the shape of their $N(E)$can roughly predict the amplitude of oscillations. Class 2 systems which are aproduct of dynamical evolution, consist of observed and simulated galaxies andclusters, and pure dark matter halos. Oscillations in the density profile slopeseem pervasive in the central regions of Class 2 systems. We argue that inthese systems, slope oscillations are an indication that a system is not fullyrelaxed. We show that these oscillations can be reproduced by smallmodifications to $N(E)$ of DARKexp. These affect a small fraction of systems'mass and are confined to $\log(r/r_{-2})\lesssim 0$. The size of thesemodifications serves as a potential diagnostic for quantifying how far a systemis from being relaxed.

Ubiquity of density slope oscillations in the central regions of galaxy and cluster-sized systems